What Is Morphological Computation? On How the …eprints.whiterose.ac.uk/109375/8/MullerWhat...

What Is MorphologicalComputation? On How theBody Contributes to Cognitionand Control

Vincent C. Müller*,**Anatolia College/ACTUniversity of Leeds

Matej Hoffmann†

Istituto Italiano di TecnologiaCzech Technical University in Prague

KeywordsBody, cognition, computation, control,embodiment, soft robotics

Abstract The contribution of the body to cognition and controlin natural and artificial agents is increasingly described as “offloadingcomputation from the brain to the body,” where the body is said toperform “morphological computation.” Our investigation of fourcharacteristic cases of morphological computation in animals androbots shows that the “offloading” perspective is misleading.Actually, the contribution of body morphology to cognition andcontrol is rarely computational, in any useful sense of the word.We thus distinguish (1) morphology that facilitates control,(2) morphology that facilitates perception, and the rare cases of(3) morphological computation proper, such as reservoir computing,where the body is actually used for computation. This resultcontributes to the understanding of the relation between embodimentand computation: The question for robot design and cognitivescience is not whether computation is offloaded to the body, but towhat extent the body facilitates cognition and control—how itcontributes to the overall orchestration of intelligent behavior.

1 Introduction

1.1 StructureIt has become increasingly common to explain the intelligent abilities of natural agents through ref-erence to their bodily structure, their morphology, and also to make extended use of this morphologyfor the engineering of intelligent abilities in artificial agents (e.g., robots). These two uses of mor-phology for explanation and engineering are sometimes referred to as “morphological computation” (seethe special 2013 issue of Artificial Life 19(1)).1 Since this notion is fairly new and fairly vague, wewill look at the existing uses, interpretations, and definitions of this concept and discuss their im-plications. The term is initially puzzling because it is sometimes used to counter the classical view

Author Contributions: The authors are listed in the sequence agreed on when this work was started some time ago. In the meantime thecontributions from the roboticist (M.H.) and the philosopher (V.C.M.) have grown to be roughly equal.* Contact author.** Anatolia College/ACT, PO Box 21021, 55510 Pylaia, Greece, and IDEA Centre, Department of Philosophy, University of Leeds, Leeds,LS2 9JT, UK. E-mail: [email protected]† iCub Facility, Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy. E-mail: [email protected]; Department ofCybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Karlovo namesti 13, 121 35 Prague 2, Czech Republic.E-mail: [email protected] This morphological computation community unites researchers from various fields, including computer science, robotics, electronics,material science, chemistry, and biology, who are in a broad sense interested in computational properties of different physical substratesor morphologies. Two editions of the International Conference on Morphological Computation were held in Venice, Italy, in 2007 and 2011,followed by the International Workshop on Soft Robotics and Morphological Computation in Ascona, Switzerland, 2013.

© 2017 Massachusetts Institute of Technology Artificial Life 23: 1–24 (2017) doi:10.1162/ARTL_a_00219

that “cognition is computation” through a study of body-dependent behavior (there it overlaps withembodiment), but then it reintroduces the notion of computation through the back door. It alsoseems to call processes and features “computational” that would not normally fall under that term—such as the morphology of geckosʼ feet that allows them to walk on smooth vertical surfaces.

In order to clear up the muddy waters of morphological computation, we categorize the conceptualsituation and come to the conclusion that morphological computation is actually extremely rare. We firstintroduce four types of characteristic cases that are often cited as examples of morphological compu-tation: (1) passive dynamic walkers, (2) self-stabilizing robots and climbing geckos, (3) insect eyes withdistribution of light-sensitive cells optimized for particular tasks, and (4) physical reservoir computing(Section 2). Then, we introduce different notions of computation, abstract and physical (Section 3), andapply them to the characteristic cases (Section 4). Based on this analysis, we classify the case studies intothree classes: morphology facilitating control, morphology facilitating perception, and morphologicalcomputation. This also serves as a “reality check” for several notions of computing (Section 5). Finally,we explain the conclusion that design and analysis should focus not on computation, but on howmorphology facilitates cognition and control—how it contributes to the orchestration of intelligentbehavior (Section 6) and how it can be specifically designed to do that—either by evolution in thecase of animals, or by engineers. A perfect example of the latter is the growing field of soft robotics.

1.2 An Initial WorryIn a classic book on the subject, Pfeifer and Bongard [61] show a flexible sprawl robot, a dog, and aconventional humanoid walking over an uneven surface (see Figure 1) and comment:

Morphological computation. (a) Sprawl robot exploiting the material properties of itslegs for rapid locomotion … thus reducing the need for computation. (b) An animalexploiting the material properties of its legs (the elastic muscle-tendon system) thus alsoreducing computation. (c) A robot built from stiff materials must apply complex control toadjust to uneven ground and will therefore be very slow. [61, p. 97, Figure 4.1 caption]

By “morphological computation” we mean that certain processes are performed bythe body that otherwise would have to be performed by the brain. [61, p. 96]

These remarks lend themselves to at least two divergent interpretations:

1. Appropriate use of the body morphology leads to a reduction of the total amount ofcomputation that is required to complete the task.

Figure 1. Morphological computation [61, p. 97, Figure 4.1].

V. C. Müller and M. Hoffmann What Is Morphological Computation?

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2. The total amount of computation is constant, but a body takes over some of thecomputation originally performed by the brain and computes it in the morphology.

In the example of walking, is this a computational task in the first place? Is the perspective of“offloading computation from the brain to the body” feasible? That is, is it a design decision thatone can take: In design (a), one decides to off-load computation to the body, and in design (c), onedecides not to, and to compute centrally in the brain/controller? Even if such a hypothetical designlandscape existed and an agent could move on it, in what sense would the contribution of the bodybe a computational one?

2 Approaching the Problem: Characteristic Cases

To clarify the picture, we will analyze four families of case studies from the morphological compu-tation literature and try to structure the diverse approaches by locating them within a conceptuallandscape: (1) the passive dynamic walker, (2) self-stabilizing machines, gecko feet, and coffeeballoon grippers, (3) the eye of the fly, and (4) physical reservoir computing.

The passive dynamic walker (1) is a classic example that is invoked in many places, but it is alsocharacteristic in that it has no control structure in the classical sense, no computer in the classicalsense, no motors, and no sensors either—a purely mechanical device. It is thus the extreme example:If this is computing, what is not? Self-stabilizing machines and gecko feet (2) can be viewed as“active extensions” of the passive walker. The physical body is complemented by actuators andsensors and their connection in a simple control loop; yet the physical interaction of the body withthe environment remains absolutely crucial for the task at hand. But the systems discussed in thecontext of morphological computation concern not only movement, but also traditionally more“cognitive” abilities like perception, and one case where this is particularly apparent, and well-studied,is the eye of the fly (3). Finally, there are cases where the whole system, the body, literally seems tocompute, so we need an example for this kind (4), which is reservoir computing. Of course, the choice ofexamples was revised in the light of our discovery that there are three distinct uses of “morphologicalcomputing” that actually talk about rather different things—so we need examples for each of theseuses. (Cases (1) and (2) belong to the same category, in our analysis.)

2.1 The Passive Dynamic Walker: Behavior by Purely Mechanical InteractionOne classical example in this field is the passive dynamic walker [52]: a minimal robot that can walkwithout any motors or control electronics. It loosely resembles a human, with two legs, a minimaltorso, and two arms, but its ability to walk is exclusively due to the downward slope of the incline onwhich it walks and the mechanical parameters of the walker (mainly leg segment lengths, massdistribution, and foot shape) (Figure 2a). The walking movement is entirely the result of finely tunedmechanics on the right kind of surface. A motivation for this research is also to show how humanwalking is possible with minimal energy use and minimal central control.

2.2 Self-Stabilizing Machines, Gecko Feet, and Coffee Balloon GrippersMost of the problems that animals or robots are faced with in the real world cannot be solved solelyby passive interaction of the physical body with the environment. Typically, active involvement bymeans of muscles/motors is required. Furthermore, the actuation pattern needs to be specified bythe agent,2 and hence a controller of some sort is required. However, it turns out that if the physicalinteraction of the body with the environment is exploited, the control program can be very simple.For example, the passive dynamic walker can be modified by adding a couple of actuators and

2 In this article, we will use “agent” for whatever acts and has some morphology, like an animal or a robot, without thus committingourselves to a particular view on what kinds of things are agents, whether they must be individuals, or the like.


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sensors and a reflex-based controller, resulting in the expansion of its ecological niche to levelground (Figure 2b) while keeping control effort and energy expenditure to a minimum [14].

While taking advantage of the natural interaction of the body with the environment can lead tovery energy-efficient locomotion, what if the agent is perturbed out of this preferred regime? Itseems that corrective action needs to be taken. However, it can be the very same mechanical systemthat can generate this corrective response. This phenomenon is known as self-stabilization and is aresult of a mechanical feedback loop. To use the dynamical systems description, certain trajectories(such as walking with a certain gait) have attracting properties, and small perturbations are automat-ically rejected.3 Blickhan et al. [5] review self-stabilizing properties of biological muscles in an articleentitled “Intelligence by Mechanics”; Koditschek et al. [42] analyze walking insects and deriveinspiration for the design of a hexapod robot with unprecedented mobility (RHex—Figure 2c; e.g.,[71]). In human locomotion, Tagaʼs seminal work dealing with entrainment among the neural system,the musculoskeletal system, and the environment has a similar spirit [76, 77], including robustnessagainst perturbations.

In some cases, a particular body morphology is the only means to achieve certain behaviors. Thisis wonderfully illustrated by the ability of geckos to climb both rough and smooth vertical surfaces,mainly thanks to van der Waals forces between their feet and the surface they are climbing on. Inorder for these forces to have sufficient magnitude, a very intimate contact between the feet and thesurface is necessary (Figure 2d). This is achieved through a hierarchical structure of compliance thatallows conformation on a centimeter scale (through flexion of body and limbs), millimeter scale(morphology of toes and lamellae on their surface), 1–50-micrometer scale (setae of which thelamellae are composed), and <500-nanometer scale (hundreds of spatulae at the setae tips) [2]. Thesefindings have inspired the design of Stickybot—a robot that can climb smooth vertical surfacesincluding glass, tile, and plastic [39]. Again, the specific ability of the gecko is the result of its mor-phology interacting with a particular environment—not primarily that of higher-level central control.

Finally, the same equally applies to other behaviors or tasks—not only to locomotion. Brownet al. [8] have devised a gripper that utilizes a unique grasping strategy. Fingers of a hand are replacedby a single mass of granular material (e.g., ground coffee). The bag containing granular material ispressed onto an object, flows around it, and conforms to its shape (Figure 2e). Then, a vacuumpump is used to evacuate air from the gripper, which makes the granular material jam and stabilize

3 The description is idealized—in reality, a walking machine would fall into the class of hybrid dynamical systems, where the notions ofattractivity and stability are more complicated.

Figure 2. Walking, climbing, and grasping creatures. (a) The Cornell passive dynamic walker with arms, which can walkcompletely passively down an incline [15]. (b) An actuated extension of the passive walker that can traverse flatground [14]. (Both pictures courtesy of Steve Collins.) (c) RHex robot [71] (picture courtesy of U. Saranli). (d) Geckofoot. (e) Coffee balloon gripper [8] (picture courtesy of John Amend).



the grasp. The gripper conforms to arbitrary shapes passively, that is, without any sensory feedback,thanks to its morphological properties only. Brown et al. identify three mechanism that contribute tothe gripping: (i) geometric constraints from interlocking between gripper and object surfaces; (ii) staticfriction from normal stresses at contact; and (iii) an additional suction effect, if the gripper membranecan seal off a portion of the objectʼs surface. The properties of the gripper can be changed by using adifferent granular material. Objects of various shapes as well as hardnesses (from steel springs to raweggs) can be gripped.

2.3 The Eye of the FlyIn previous subsections, we have outlined a few characteristic cases that illustrate the key role theparticular body morphology plays in performing certain behavior. The focus was thus on the impli-cations of the shape and material properties of the body for direct physical interaction with theenvironment. However, the body morphology critically shapes the information flow in an agentas well. In particular, the type of sensory receptors—their mechanism of transduction—determineswhat kind of signals the agentʼs brain or controller will be receiving from the environment. Further-more, the shape and placement of these sensors will perform an additional transformation of theinformation that is available in the environment.

For example, different species of insects have evolved different non-homogeneous arrangementsof the light-sensitive cells in their eyes, providing an advantageous nonlinear transformation of theinput for a particular task. One example is exploiting ego-motion together with motion parallax togauge distance to objects in the environment and eventually facilitate obstacle avoidance. However,to correctly compute the distance to the objects, the sine of the angle spanned by the objectʼs mo-tion on the eye is involved (see Figure 6 in [25]), which would have to be taken into account by themotion detection circuitry—if the spacing of light-sensitive cells were uniform. However, the dis-tribution of the cells is nonuniform and follows a sine gradient in the interommatidial angle, suchthat sampling of the visual space is finer towards the front than laterally (the head of a male houseflycan be seen in Figure 3a). This effectively compensates for the sine relationship in the formula andallows uniform motion detection circuitry to be used everywhere. Franceschini et al. [25] performedthis analysis on a housefly and at the same time designed a mobile robot that avoids obstacles usingthe same concept. Recent designs of artificial eyes with design inspired by arthropods include [73]and [23] (Figure 3b).

2.4 Physical Reservoir ComputingUnlike typical robot bodies that often have rather simple geometrical forms and are composed ofrigid materials, biological bodies typically have highly complex shapes and are soft and deformable.These properties also make their dynamics much richer. This can be exploited for directly achievingparticular behavior in the physical world on one hand, and on the other hand can be utilized forcomputational tasks.

Figure 3. Eye morphology. (a) Eye of a fly. (b) CurvACE artificial compound eye [23] (image courtesy of Dario Floreano).



The neural networks community has recently proposed a family of architectures that has beennamed reservoir computing (see, e.g., [47] for a review). There is a large collection of neurons with non-linear activation functions and with recurrent connections that have a random but bounded strength;this is referred to as a dynamic reservoir. These neurons are randomly connected to input streams, andthe dynamics of the input is then spread around and transformed in the reservoir, where it resonates(or “echoes”—hence the term “echo-state networks”) for some time. It turns out that tapping intothe reservoir with simple output connections is often sufficient to obtain complex mappings of inputstream to output stream that can approximate the input-output behavior of highly complex nonlineardynamical systems (e.g., [37]). During training, the weights from the input streams and between thereservoir neurons are left intact; only the output weights—from the reservoir to the output layer—aremodified by a learning algorithm (e.g., linear regression). The complexity of the training task has beengreatly reduced (as opposed to training all the connections—see [36] for details) by exploiting thereservoir to perform a spatiotemporal transformation of the input stream (the temporal aspect ofthe input sequence has basically been unfolded by the reservoir and can be retrieved directly atany instant). Furthermore, if feedback loops from the output back to the reservoir are introducedand subject to training, the network can be trained to generate desired output streams autonomously.

Interestingly, a physical device can act as a reservoir as well. This can be simply a bucket of water[22] or the body of an agent. Biological bodies interacting with their environments tend to possessthe properties required—nonlinearity and fading memory—and might thus be employed as spatio-temporal filters. Hauser et al. [27, 28] provide a theoretical foundation for these mechanisms:

The underlying idea is to view the morphological structure as some fixed nonlinearkernel, which provides us with high-dimensional projections and nonlinear combinationsof our input. Hence, the required nonlinearity (next to the dynamics) is provided bythe morphological structure itself and, therefore, linear feedbacks and readouts aresufficient in order to emulate nonlinear differential equations. [28, p. 601]

Concrete demonstrations of this property are provided using simulated mass-spring systems. Forexample, different networks consisting of dozens of masses and springs with linear feedback loops(see Figure 4) are trained to emulate different nonlinear limit cycles (Van der Pol oscillator, quadraticlimit cycle, Lissajous figure) as well as four output streams corresponding to motor patterns for fourquadruped gaits. Johnson et al. [38] have demonstrated that a similar mass-spring system can be evolvedto replace a continuous-time recurrent neural network in the famous categorization task of Beer [4].

This idea has been adopted by others and demonstrated in other systems too. Caluwaerts et al.[10] have explored this approach in a simulated “robot”—a tensegrity structure consisting of fixedbars and passive and active springs. The body was employed as a reservoir to sense the properties ofdifferent terrains. Then, a simple readout mechanism was trained to classify the grounds. Further-more, the tensegrity structure, accompanied by a feedback loop and a readout mechanism, was also

Figure 4. Spring-mass computer [28]. (Image courtesy of Helmut Hauser.)



shown to generate a periodic pattern similar to one produced by a nonlinear oscillator [11]. Thisoutput could then be used as a command sequence sent to actuators. However, this approachwas further complicated in a quadrupedal robot [19]. Nakajima et al. [57] have used a model of asoft robotic arm inspired by the octopus. In a simulated arm in an underwater environment, theysuccessfully embedded the same benchmarks used by Hauser et al. in the open-loop system (2nd-and 10th-order dynamical system and Volterra series [27]) and in the system with feedback (Van derPol equations, quadratic limit cycle, and Lissajous curve [28]). Extensions on an actual physicalsilicone arm in a water tank were presented in [58].

3 The Relevant Notions of Computation: Abstract and Physical

Before we try to analyze the characteristic cases with respect to their relationship to computation, wewill set the ground for this exercise by reviewing different notions of computation and, in particular,what it means for a physical system to compute.

3.1 Abstract Computation and Its ModelsWe will briefly introduce the most important models of abstract or mathematical computation. Thefirst will be the classical Turing model. Second, we will introduce a class of “nonstandard” or“natural” computation models that try to go beyond the Turing model and lift some of its constraints.

3.1.1 Digital Turing ComputationThe classical notion of computation starts from a digitally encoded input, which is processed step bystep, following an algorithm, giving rise to a digital output. One formalization of this notion is theTuring machine—an abstract device that manipulates symbols on a tape according to a table of rules.The computation is serial and batch, that is, the computation is finished and complete output is pro-vided only once the machine halts. No additional inputs can be introduced during the computation.This can be used to define the family of effectively (or algorithmically) computable functions, asoutlined by the Church-Turing thesis:

All and only the effectively computable functions can be computed by a Turing machine.[13, 80; also see 55]

Alternative, but equivalent, formalizations are possible using lambda calculus [13], recursivefunctions [40], or other techniques.

To move from one digital state to another in a Turing machine is a step-by-step process thatfollows a finite rule, that is, an algorithm, which is itself formal. So, the computation is a purelysyntactic process that can be realized in several different ways on a physical device: It is thus multiplyrealizable, and since the states are digital, it is exactly multiply realizable.

3.1.2 Natural ComputationThe Turing model of computation has been extremely powerful and is sometimes regarded as syn-onymous with computation itself. However, some of the features of the Turing model (digital,batch, serial, etc.) can also be constraining, especially with regard to realizing the computation onsubstrates that are common in nature. “Natural computing” is an umbrella term that encompassesefforts to characterize computations performed by natural systems—neural computation, forexample—as well as realizing computation on alternative hardware, like molecular (or DNA) com-puting or quantum computing.

Some classes of problems, like the continuous domain, can only be approximated by Turingmachines; some, like parallel asynchronous computing, lie outside the Turing model altogether.



However, to qualify as computation, an alternative model is required that provides a formal descrip-tion of the states and their evolution.

One possible formalism for analogue computing is provided by dynamical systems theory: Adynamical system is governed by its internal dynamics, and it may also receive a stream of inputsignals and produce a stream of output signals.4 Crutchfield et al. [16] provide a broader contextfor computation by complex systems in their introduction to a focus issue of the journal Chaos :“Information Processing in Dynamical Systems—Beyond the Digital Hegemony.” Dambre et al.[18] have studied these real-time mappings and defined the information-processing capacity of adynamical system and how it can be inferred, allowing for a comparison of computational propertiesof a broad class of dynamical systems. Time-invariant systems with fading memory—which oftenserve as a formalism in reservoir computing (including physical reservoir computing [27])—are oneparticular class of these systems. Maass et al. [49] present a computational model based on transientdynamics in high-dimensional neural circuits. Unlike in the Turing model, no sequential transitionsbetween discrete states are necessary, yet universal computational power can be achieved.5

In theory, natural computation would be a superset of Turing computation, but it is rather acollection of different possible computational models that lift some of the constraining featuresof the Turing model (digital, batch, serial, etc.; see [20] for a detailed account). An analogue com-puter uses continuously changing processes and requires some sort of input, a process, and then ameasurement of the outcome to a certain degree of accuracy. A classic example is the slide rule; aless mathematical example is the bombsights used in WWII bomber aircraft to determine the op-timal time of bomb release to hit a target, given the angle, altitude, and wind speed (taking intoaccount gravity and air drag). Both were replaced by digital computers in the 1960s. If the compu-tational model is analogue, it is not possible to realize it exactly in physical machines, and it is thusonly approximately multiply realizable.

3.2 When Does a Physical System Compute?The models of computation discussed above are solely on the abstract level; for computation to beperformed in the real world, they need to be realized on a physical system. A prime example is thestandard digital computer, which comes practically close to realizing the (abstract) Turing machine inan electronic device (only the computerʼs memory is not unbounded like the tape of the Turingmachine). This particular physical realization is unproblematic, and it naturally occurs to us thatthe digital computer performs computation. However, the situation will become more complicatedas we discuss whether other physical systems are computing.

3.2.1 Designed, Useful Computation with Encoding and DecodingA concrete formal framework addressing “When does a physical system compute?” was recently putforth by Horsman et al. [32] in an article with that title. They tackle the question as a relationship be-tween an abstract object (a computation) and a physical object (a computer), which is best explained byusing a schematic (Figure 5) to illustrate the correspondences. These correspondences are encapsulatedin a theory T. There are then two principal perspectives one can take. The first one is that of anexperimental science, like physics, that aims at understanding a (physical) system. This is a modelingendeavor—a theory is sought that allows describing the evolution of the physical system in abstractterms, and it can be tested using experiments. That is, under the particular theory, the state of thephysical system, p, is observed and turned into its abstract counterpart, mp, using the representationrelation RT. Then the physical system is left to evolve, and in parallel the abstract layer is also evolved,using the abstract dynamics, CT, which should parallel the physical dynamics,H. Finally, if the theory is

4 A possible bridge between dynamical systems and digital computation is the symbolic dynamics framework (e.g., [45]).5 This architecture, that of liquid state machines, was later united with echo-state networks under the “reservoir computing” umbrella.



valid, the abstract final state, mp0 , should be “close enough” to the representation of the physical

final state, RT (p0), that is, mp 0. The abstract and the physical layer should thus “commute,” and

the theory can be used to predict the evolution of the physical system (running only the path fromp over mp to mp

0 ).The other perspective is essentially complementary and involves using the physical system to predict

the outcome of an abstract evolution. It relies on the quality of the theory T that produces a commutingdiagram, but requires in addition a representation relation in the opposite direction—the instantiation IT.That is, it has to be possible to find a physical object corresponding to an abstract description (encodingthe abstract state—e.g., encoding a “1” as voltage high). Here we are moving from the realm of science(modeling) to the realm of technology—the theory can be used to construct (engineer) a physical sys-tem in a given state. With that in place, the physical system can finally be used as a computer using thefollowing trajectory: An abstract initial state mp is encoded into the physical system using IT (mp), givingrise to p; the physical system is left to evolve, reaching p0, which is finally decoded using RT(p

0), givingrise to mp 0, which we are assuming corresponds to mp

0—the desired outcome of the abstract evolu-tion. In this case, the physical system has acted as a computer.

Horsman et al. [32] conclude their framework by stating the necessary requirements for a physical systemto be capable of being used as a computer :

a. A theory T of the physical computational device that has been tested in relevant situationsand about which we are confident.

b. A representation {RT, IT} of the physical system that is used for representing the initialstate of the physical system (encoding using IT) and also for the final state, so that outputis produced from the computation (decoding using RT).

c. At least one fundamental physical computational operation that takes input states tooutput states.

d. The theory, representation, and fundamental operation(s) satisfy the relevant sequence ofcommuting diagrams.

This description is agnostic with respect to the particular abstract computation type (digital Turing ornatural) as well as to the nature of the physical system—the important thing is that the theory issufficiently advanced and reliable that we can rely on the commuting diagrams in all scenarios we areplanning to use the computational device in. The advantage of a digital computer is that a perfect,rather than a “close enough,” match between mp

0 and mp 0 is guaranteed.

Figure 5. Full commuting diagram between an abstract and a physical system. The relationship between the abstract andthe physical system is established by the theory T. The dynamics of the abstract system (transition between initial andfinal states) is described by CT. The initial state of the abstract system is denoted as mp, and its final state as mp

0 . Thephysical systemʼs dynamics isH, and its initial state and final state are denoted as p and p0, respectively. RT is the modelingrelation that maps physical entities p to their abstract models mp (also decoding). IT is the inverse instantiation relation(encoding) that goes in the opposite direction and allows one to physically instantiate an abstract entity. (Created basedon multiple schematics from [32].)



3.2.2 Without the User, There Is No ComputerThere is an additional important aspect that is implied by condition b—representation, involvingencoding and decoding. As Horsman et al. put it:

A necessary condition of there being representation present is that there is, as well asthe computer, an entity capable of establishing a representation relation. That is, anentity that represents this specific physical system as this specific abstract object,encoding and decoding data into it. Something must always be present that is capableof encoding and decoding: if there is a computer, what is using it? [32, p. 15]

This definition matches with the conditions for computation presented by Crutchfield et al. [16, p. 2],who characterize systems performing “designed computation” as machines that “process information inways that are useful to us”—and it is one approach to the symbol grounding problem [54]. In the samevein, Piccinini [66, p. 740] introduces a “Usability Constraint: if a physical process is a computation, itcan be used by a finite observer to obtain the desired values of a function”—so he allows for a computerthat is not actually used—as does the “programmability” proposal in Zenil [84].

Finally, another definition of computation—specifically targeted to encompass morphologicalcomputation—was developed for the 2007 International Conference on Morphological Computationunder the leadership of Norman Packard (cited from the call for papers of the 2011 conference),defining a process to be a computation if:

(1) we can identify relation between input and output systematically,

(2) the process is programmable for the generation of different classes of non-trivial input-output mappings, and

(3) it is useful for some specific purposes.

Again, conditions (2) and (3) seem to step out of the computational process itself and take an outsideperspective of an observer or user of the computation. That is, the mappings need to be programmable(by someone) and useful ( for someone ). This observer or user can presumably be the agent itself, or it can behumans that exploit a particular physical system to perform a specific type of mapping that is useful tothem. The encoding/decoding step is not explicitly addressed here, however.

3.2.3 Intrinsic ComputationThe world is naturally populated with a variety of physical systems of different properties and scales.Many of them can be said to have some computational properties (like storing and generating in-formation) without being “designed computers,” so they “intrinsically compute” [16]. Information-theoretic methods can be used to measure the amount of computation (see, e.g., [83] for an examplein quantum systems). Some physical systems will then have richer or more interesting computationalproperties than others—for example, they can be very fast or compute in parallel. Specificallyrelevant to morphological computation, biological bodies interacting with their environments tendto possess nonlinearity and fading memory, which can be exploited by the reservoir computingframework (Section 2.4).

If there is a user, encoding and decoding are designed, and the physical system is sufficientlyreliable, then the natural computing substrates can become computers according to the frameworkpresented in the previous section. However, Horsman et al. [32] point out that this computational(abstract) description of the physical evolution should be available a priori. Only then can thephysical system be used to predict anything (the evolution of an abstract system). Conversely:

A common, and unfortunate, method of ascribing computational ability to a non-standardsystem is as follows. A novel computing substrate is proposed (a stone, a soap bubble,



a large interacting condensed-matter system, etc.). The physical substrate is “set going”and an evolution occurs. At a certain point, the end of the process is declared andmeasurements taken. The initial and final states of the system are compared, and thena computation and a representation picked such that if the initial state and final statesare represented in such a way, then such a computation would abstractly connect them.The system is then declared to have performed such a computation: the stone hasevaluated the gravitational constant, the soap bubble has solved a complex optimizationproblem and so on.… If a computational description of a physical evolution can onlybe applied post-hoc, then the system has not acted as a computer. [32, p. 20]

They argue that numerous representations can be invented post hoc, and hence this approach may lead toa special type of pancomputationalism (“everything computes”—see Section 3.2.4 below). Intrinsic com-putation is thus loosely defined, encompassing all physical systems with some “interesting” computationalproperties. We have depicted this schematically in Figure 6 as a superset of designed computation.

3.2.4 Pancomputationalism: Everything Is a ComputerThe position that the universe is a computer, that all processes are computation, is now often called“pancomputationalism,” a term probably first used by [65] and [24, p. 566]. In its strongest form,pancomputationalism claims that the universe is literally a computer that computes the changes ofthe universe according to the physical laws; that all processes are computational processes. Our con-ventional computing machines would thus be only a tiny fraction of all the computing going on. Thisis really a cosmological thesis of physics and has its origins in that discipline. For a discussion andrecent literature, see [21, esp. Section 2; 56]. Pancomputationalism implies that cognitive and mor-phological processes are trivially computational in the sense that everything else is.

In the morphological computation literature, the pancomputationalist dissolution looms in re-marks like “Control implies computation” (Füchslin et al. [26, p. 10]). If one takes this literally, thenno difference between computational and other control can be made any more. But we normallywant to say that, for example, the centrifugal governor in a steam engine is a dynamic system andprovides proportional control. The centrifugal governor is not a computer, even if it can be replaced

Figure 6. Different notions of computation performed by a physical system. Designed computation refers to physicalsystems engineered in such a way that their physical evolution can be used to predict the outcome of an abstract evo-lution (see Section 3.2.1). Intrinsic computation lacks a proper definition, but refers to natural physical systems withinteresting computational properties (Section 3.2.3). Pancomputationalism spans the whole hypothetical space, claimingthat everything computes (Section 3.2.4). Finally, “offloaded computation” is defined negatively as computation that doesnot have to be performed by a brain/controller (Section 3.2.5).



by a computational system that measures the relevant values, computes a response, and controls therelevant value (valve opening). If the remark “this is a computational process” deteriorates into “thisis a process,” then it becomes pointless. In Figure 6, this is simply illustrated by pancomputation-alism being the ultimate superset.

3.2.5 Morphological Computation as Offloading from the BrainA notion of computation that is very popular among the morphological computation communityseems to rely on a negative or complementary definition of morphological computation. This iscomputation in the sense of offloading from brain to body (e.g., [61, p. 96]) that we introducedin Section 1.2. Another instance appears in Hauser et al. [27, p. 356]:

A rigorous implementation of this concept [morphological computation] are passive walkers.The first of a series was developed by McGeer [52]. Typically, such a robot has no activecontroller at all. Only its passive physical structure maintains the balance in a robust fashion,while it walks down a slope. Therefore, one could argue that the computation, which isneeded in order to balance the robot robustly, is “computed” by the physical body itself.

The offloading argument is also picked up in Füchslin et al. [26] and illustrated in Figure 1 there. In-terestingly, the schematic matches very well with our Figure 5. Therefore, we have adapted our Figure 5to match the one from [26], resulting in our Figure 7. The goal is to bring a physical system (e.g., body ofan agent) from initial state p to final state p0. This is achieved jointly by the physical dynamicsH and bythe action of the controller that processes the sensory input to obtain a representation mp of the physicalsystemʼs state p. With a goal state mp

0 , in mind, it computes a control action that should bring about thedesired transition and translates this into actuator commands. At the same time, the physical system issubject to its own evolution (dynamics), H. In passive walkers, the abstract level is completely absent;there is only physical dynamics. Conversely, traditional robotics tends to suppress “body dynamics” or“natural dynamics” (see, e.g., [33] or [12] for a formal definition) in favor of bringing about the desiredstate transitions by means of the computed control action. The idea of “morphological control” is toexploit the physical dynamics to the maximum, while at the same time reducing the contribution of thecontroller (the abstract-level loop)—as illustrated by the bold arrowH(p) in the schematic.6 This is theninterpreted as offloading computation from the brain/controller to the body.

It may be possible to examine the amount of computation on the controller side, and differentcontrollers can then be compared. This exercise becomes more interesting if one is allowed to mod-ify also the physical system (the agentʼs body), although this then becomes much harder to compareformally. Rückert and Neumann [70] studied learning of optimal control policies for a simulatedfour-link pendulum that needs to maintain balance in the presence of disturbances. The morphology(link lengths and joint friction and stiffness) was manipulated, and controllers were learned for everynew morphology. They showed that: (1) for a single controller, the complexity of the control (asmeasured by the variance of control gains of a time-varying linear feedback controller ) varies withthe properties of the morphology: certain morphologies can be controlled with simple controllers;(2) optimal morphology depends on the controller used; (3) more complex (time-varying) controllersachieve much higher performance than simple control across morphologies.

Among those conclusions of Rückert and Neumann [70], (1) could be interpreted in the“offloading” spirit, but the performance really depends on a complex interplay of the controller,body, and environment, and the offloading or outsourcing will thus practically be very complexand hard to express in a formal framework. More importantly, this scenario allows us to understandwhether the contribution of the body is computational in the sense of the notions presented above.

6 Some schemes from control theory with appropriate cost functions will automatically result in minimal control actions and thus“optimize the contribution of the morphology.” For example, Moore et al. [53] used discrete mechanics and optimal control to steera satellite while exploiting its dynamics to the maximum.



We claim that it is clearly not, in any sense but the pancomputationalist one. Adopting the framework of“designed, useful computation with encoding and decoding” (Section 3.2.1), we see that the physicalsystem is definitely not used to predict any abstract evolution. Instead, the goal is in the physical space(such as walking), and the abstract level may only serve to help achieve this in the form of a controller (infact, the abstract computation performed by the controller will have to run on some hardware too, butthis will normally not be the agentʼs body that is walking, but a neural circuit or a digital chip). We are notinterested in computational properties of the physical system either (what the passive walker or thependulum is intrinsically computing). Therefore, the label “morphological computation” is misleading,as the body is not computing in any but a metaphorical sense of taking away some load from the brain—perhaps the label “morphological control” might be appropriate [26].

Thus, faithful to the notion of a complement to brain computation, one can illustrate this off-loaded computation schematically in Figure 6 by interpreting the physical computation types wehave listed so far as sets. Then, if one assumes that there is a task (say walking) and that all processesare computational (pancomputationalism), the amount of computation needed for the task forms auniverse P. Then, if one knows the amount of computation performed by the central controller, sayC, then the morphological computation in the outsourced sense could be the absolute complementof the set that represents computation performed by the central controller, that is, P excluding C.The offloaded computation, O, would be equal to

O ¼ Cð ÞC ¼ P∖C:

This is schematically illustrated in Figure 6.

3.3 Computing ConcludedWe have reviewed different types of abstract computational models as well as a number of perspectiveson when a physical system computes. For our purposes—assessing when physical bodies act as com-puters—it is the latter that is key. To this end, we find the first notion presented, “designed, usefulcomputation with encoding and decoding” (Section 3.2.1), the most concrete and strict and also themost useful one. The second one, “intrinsic computation” (Section 3.2.3) is more problematic, but if asystem is found to have “interesting intrinsic computational properties,” it can be further inspected (ifitʼs a natural system) or modified (if itʼs artificial) to see if the stronger requirements—encoding, decod-ing, user—can be established. Paraphrasing Horsman et al. [32, p. 15], this allows us to distinguishbetween a physical system “going about its business” while potentially being a computer, and asystem whose physical evolution is used to compute. The pancomputationalist notion of computation(Section 3.2.4) does not seem to be helpful to our analysis. Finally, the computation in the sense ofoffloading from brain to body (Section 3.2.5) seems to be also a problematic perspective that is hardto defend beyond the level of metaphor.

Figure 7. Exploiting morphology for control. See text for details, and also Figure 5 for the definition of symbols.



4 A Look at the Characteristic Cases: Computation?

Let us now look back at the characteristic cases that we introduced in Section 2 in the light of thedifferent notions of computation. We summarize our findings in Figure 8.

4.1 The Passive Dynamic WalkerAlthough the passive dynamic walker (introduced in Section 2.1) is often taken as a prominentexample of morphological computation, it really is only “pure physics walking” or a “physical systemgoing about its business.” That is, it definitely does not satisfy the necessary conditions for“designed, useful computation with encoding and decoding” (Section 3.2.1). There is no abstractcomputation that it was designed to predict through its physical evolution (walking down the slope).Also, although McGeer [52] does offer a model of the walker—its equations of motion—it wouldhardly qualify as a system with particularly interesting computational properties and as thus comput-ing intrinsically (Section 3.2.3). Therefore, it is hard to conceive of the walker as computing in anybut the pancomputationalist sense, that is, the sense that all physics is computing. Finally, is thewalker doing computation that was offloaded from the brain? Hardly, as there is no brain the com-putation of which could be offloaded. At the most, the hypothetical computation that “is needed”for walking has been fully offloaded from a hypothetical controller to the morphology of the walker.Yet, this offloading landscape is purely metaphorical. To illustrate this graphically, we have simplylocated the passive walker inside the pancomputationalist universe in Figure 8, possibly qualifying asthe offloaded computation showcase, but, again, only in a metaphorical sense.

4.2 Creatures Exploiting Morphology to MoveA similar line of argumentation holds for the other examples introduced in Section 2.2. The activedescendants of the passive dynamic walkers (Figure 2b), the self-stabilizing robots (e.g., RHex—Figure 2c), the gecko (Figure 2d), or the coffee balloon gripper (Figure 2e) do not seem to be com-puting in any but the trivial, pancomputationalist, sense either. They may be more useful than thepassive dynamic walker, but it is directly their physical behavior in the environment that is useful—there is again no abstract computational level that would, under some encoding and decoding,“commute” with the physical level. All of them have a brain or controller of some sort, so the

Figure 8. Characteristic cases and the notion of computation involved. (a) Cornell passive dynamic walker [15].(b) RHex robot [71]. (c) Gecko foot. (d) Coffee balloon gripper [8]. (e) Eye of the fly. (f) Artificial compoundeye [23]. (g) Spring-mass computer [28].



offloading thought experiment is somewhat facilitated. One can imagine this in the RHex robot,for example: If the body had a different design, a more complicated controller might be neededto stabilize the robot during locomotion. However, the gecko feet seem to further compromise theoffloading perspective—if the gecko did not have feet with the particular morphology, it would simplyfall off the vertical surface, no matter how big a brain was available. In conclusion, we think that allthese are examples of morphology facilitating control and that a computational perspective is misleading.We will devote a separate subsection (Section 5.1) to these cases.

4.3 The Insect Eye PreprocessorUnlike in the previous cases, a computational perspective seems better applicable to the eye of a fly(Section 2.3). The eye morphology performs a nonlinear transformation of the visual input thatgreatly facilitates subsequent processing. Does this case fit the framework of Horsman et al.[32]—Figure 5? Not really. The problem that needs to be solved by the agent is not solely anabstract one, but lies at the interface of the physical world, which needs to be sensed, and theabstract world, where the agent needs to estimate the distances to objects.7 That is, the agent doesnot start from an abstract problem it would need to solve by encoding it into a physical system andthen reading it out again after the physical systemʼs evolution. Instead, objects in the physical worldare transformed into activations of light-sensitive cells and later into neural activations (“decoded”from the physical to the abstract system in the sense of Figure 5—this corresponding “encoding”according to standard usage in nueroscience though). After that, this information is processedby the agentʼs brain to generate an appropriate motor response (obstacle avoidance). It is asensory-motor loop that brings about control of the flying agent, and it is the first part of the controlloop where the morphology steps in. There, part of the transformation that needs to be accomplished iscarried out by the morphology itself, leaving less work to the neural circuitry after processing of thesensation takes place. One could even say that evolution has “designed” the morphology to computethis function. Nevertheless, the whole loop does not match with the one that corresponds to using aphysical system to predict abstract evolution, as required by the strict definition presented in Section3.2.1. On the other hand, one could say that the eye morphology has interesting computational prop-erties (equivalent to a function involving a sine term), which are “intrinsically” performed by the mor-phology. This is illustrated schematically in Figure 8.

4.4 Computing with Masses and Springs, Humanlike Bodies, and Octopus Arms?Buckets of water, masses and springs, tensegrities, and octopus arms were introduced earlier(Section 2.4). Interestingly, they all satisfy the properties that are necessary to act as a reservoir: highdimensionality, nonlinearity, and fading memory. They can thus be employed as a nonlinear kernel toperform a spatiotemporal transformation of incoming data. In a second stage, using a readout mech-anism implemented in a “classical” computer, they give rise to a powerful information-processingpipeline, which is particularly suited for some tasks, like classification and prediction of time series.In a biological context, perception is a prime example where classification and prediction of incom-ing input streams is highly relevant. In this sense, the biological reservoirs—such as soft bodies—perform a similar function to that of the eye morphology from the previous section. First, however,they add the temporal dimension to the sensory preprocessing they can perform. Second, unlike theinsect eye, which is specialized in performing one particular transformation, the spatiotemporal un-folding of the input stream performed by the reservoir is more universal, and there can be severalreadout mechanisms that take advantage of it.

The computational capabilities of a reservoir can be further increased if the loop is closed by feedingthe reservoirʼs output back to it. Then, it is possible to autonomously generate a broad class of limitcycles, such as the Van der Pol oscillator, quadratic limit cycle, and Lissajous figures [28]. Importantly,

7 This division—abstract versus physical—is quite arbitrary, though. In the organism, this sensing task is part of a closed loop that isconnected to the motor neurons in the flyʼs wings that directly bring about the obstacle avoidance.



these have been demonstrated in two different physical (simulated) reservoir computing setups: amass-spring system [28] and an octopus arm [57].

With this additional step, it seems that the reservoir is truly used as a computer—it will compute theevolution of an abstract mathematical system, such as a nonlinear oscillator. The model of abstractcomputation is likely to be one of the natural computation class (Section 3.1.2). For example, Hauseret al. [28, p. 356] write: “Regarding the type of computation, we consider mathematical models, whichcan be characterized as complex mappings of input to output streams in continuous time.” In thiscase, we are thus for the first time dealing with a proper computer in the sense of the requirements of“designed, useful computation with encoding and decoding” (Section 3.2.1) (hence its placement inFigure 8), with perhaps one reservation: Although the bodies can be recruited for computation, onecan hardly say that they were “designed” with this goal in mind. Furthermore, since different dynam-ical systems benchmarks were instantiated in different reservoir systems, there is even an importantadditional feature of computation—that of universal computation. It thus follows automatically that thesesystems would also be intrinsically computing and they are naturally also compatible with the offload-ing perspective. In fact, taking the case of biological bodies, one could perhaps say that an additionalcomputing substrate is instantiated in the body where the brain could outsource its tasks.

5 Discussion: Three Roles of Morphology

In our survey of the case studies, we have come across different instances of morphological compu-tation and we have examined to what extent they satisfy their respective requirements. Confronting thecharacteristic cases with these conditions gave rise to their approximate localization on the landscape ofthe computational notions (Figure 8). Interestingly, many ended up on the outer circle, far from theimaginary computational core at the center of the schema. In light of this analysis, we propose a simpleclassification of the case studies into three types:

A. Morphology facilitating control

B. Morphology facilitating perception

C. Morphological computation

This classification, along with some representatives for each category, is depicted schematically in Fig-ure 9. The true computational character of the deployment of morphology increases from left to right.

5.1 Morphology Facilitating ControlThis category could also be called “morphological control” [26], “mechanical control,” or, in a narrower,open-loop sense, “morphology facilitating actuation.” The passive dynamic walker, the creatures ex-ploiting mechanical self-stabilization, the gecko relying on its special feet, and the jamming-based coffeeballoon gripper would all fall into this category (Figure 9a–d). Also, all the examples of locomotion(quadruped Puppy and fish Wanda) and grasping (Yokoi hand) cited in [63] belong here. Other fittingrepresentatives would be the pneumatic actuators embedded in elastomers that can move and grip [35],and the switching of behaviors exploiting attractors in the mechanical system in [59]. In all these casesthe body is ingeniously contributing to the task; it enables physical behavior in the real world. However,as we argued in Sections 4.1 and 4.2, it is hard to conceive of the systems as computers in any but thepancomputationalist sense. The offloading computation perspective is also problematic, since, first, thepossibility of shifting tasks between the controller and the body is practically limited; second, even ifthrough better exploitation of body morphology the controller can be simplified (and that can be quan-tified), that does not imply that the body has taken over any computation. In fact, calling it computationmay allude to a much stronger thesis, namely that what is done by the interaction of the body with theenvironment here could also be done by a conventional computer. And that is quite the opposite ofwhat most proponents of the morphological stance want to say, since they want to stress the importanceof embodiment, in fact its indispensability. Therefore, our claim is that exploiting morphology for



control should be better handled in a non-computational context. This argument is elaborated in [30]along with a discussion of the implications of simple versus complex bodies.

5.2 Morphology Facilitating PerceptionThe eye-of-the-fly case study (Sections 2.3 and 4.3) is much more compatible with a computationalviewpoint. The transformation performed by the insect eye is a well-defined input-output mapping thatis useful for the agent and provides input to central processing. In this case, the morphology performsan important preprocessing step—takes over part of the transformation that would otherwise have tobe performed by the brain at a later stage. Other examples that might fall into this category are the non-homogeneous distribution of photoreceptors in the human eye (optimized for acuity rather than motiondetection), the mechanotransduction in the cochlea of the ear [50], and the sensilla morphology incrayfish [75]. Recent examples from robotics include strain-vector-aided sensorization of soft structures[17] and sensing through compliant artificial pneumatic muscles [34, 74]. However, the morphology (thephysical system) is not deployed here as a computational substrate to calculate an abstract problem—this would involve encoding the abstract entity into the morphology and eventual decoding. Instead,the morphology acts at the interface between physical quantities, and their representation in the brain/controller of the agent. At this interface, they introduce advantageous transformations that facilitatesubsequent processing. Obviously, the morphology has no ambitions regarding multipurpose or uni-versal mappings; on the contrary, it is highly specialized for one particular transformation.

Figure 9. Three roles of morphology. The passive dynamic walker (a) [15], the robot RHex (b) [71], the gecko foot (c),and the coffee balloon gripper (d) [8] are featured as representatives of morphology facilitating control. The eye of the fly(e), artificial compound eye (f) [23], cochlea (g), and whiskers of a rat (h) are examples of morphology facilitatingperception. Morphological computation is represented by the spring-mass computer [28] and the octopus arm reservoir[58]. (Picture G is from Wikimedia Commons; picture H by Dawn Huczek, Wikimedia Commons; picture J courtesy ofKohei Nakajima.)



Of course, the preprocessing performed by the morphology tells only half of the story—a passiveperception one in this case. Organisms interact with their environments in a closed-loop fashion:Sensory inputs are transformed into motor outputs, which in turn determine what is sensed next.Therefore, additional advantageous transformations can be achieved if active perception (e.g., [3]) is con-sidered. Active touch constitutes a perfect example where the sensor morphology, such as the length,placement, and mechanical properties of a ratʼs vibrissae (whiskers), is as important as the whiskingstrategy—active exploration of the environment (see [67] for a survey). These findings can also be trans-lated into artificial systems, such as whiskerlike sensors [68, 64]. Lungarella and Sporns [48] demonstratehow sensory morphology and closed-loop interaction can be quantified using information-theoreticmeasures. An example on this theme, where sensor morphology and active sensing go hand in handand biological inspiration finds its way to robotic systems, is underwater electrical sensing [6, 7]. How-ever, for the purposes of this article, the effect of sensing actively cannot be directly attributed to themorphology anymore and thus has remained essentially out of our scope here.

Four representatives of this category are pictured in Figure 9e–h.

5.3 Morphological ComputationFinally, the only case studies that came close to truly computing were the physical reservoir com-puting cases (Sections 2.4 and 4.4). Employing the reservoir computing framework, systems thathave properties akin to human or animal bodies can be shown to have interesting capabilities inperforming advantageous—and quite general—spatiotemporal transformations of incoming timeseries. Moreover, they are capable of autonomously emulating a broad class of dynamical systems.As this has been demonstrated in multiple systems (masses and springs, octopus arm), these are, inour view, demonstrations of true computation that can be performed by humanlike morphology.

Let us note here that the reservoir computing approach has also been successfully implementedin optoelectronic setups [44, 60]. Nonlinear optoelectronic oscillators subject to delayed feedbackhave been used to perform digit recognition and time series prediction [44] and speech recognitionand nonlinear channel equalization [60], using the reservoir computing concept. Furthermore, a firstdesign for an analogue readout mechanism in an optoelectronic setting has been presented [72],giving rise to a fully analogue setup. However, the optoelectronic implementations have a solelycomputational goal—they are basically an alternative substrate, in which the combination of physicalproperties with the reservoir idea gives rise to a practically very powerful computer. Thus, theywould fall into the realm of computation performed in alternative hardware, such as molecular orquantum computing, which really is full-blown computer technology where the morphology isdesigned to be transparent with respect to the computation; such systems have remained largely outsideof our scope. These setups work on a completely different scale and speed than the macroscopic human-body-like reservoirs mentioned earlier, which grants them higher bandwidths.

On the other hand, the macroscopic systems—such as nonlinear compliant bodies of human size—have intrinsic limitations with respect to computational tasks. In particular, they are very slow and noisyand they have to perform other tasks at the same time. Yet, they are there for free, so to speak, so theagent may consider employing their computational resources for some tasks, and reservoir computingcan provide the framework. This very much blurs the boundaries between the brain as the seat of com-puting and the “mere physical body,” in accordance with biological reality, which will be discussed in thenext section. However, thus far the examples of this type [28, 57, 58, 11, 19, 37] have a theoretical orproof-of-concept character, and their applicability remains to be proven (Figure 9I–J).

6 Applications and Conclusion

6.1 Biological Reality Blurs the BoundariesWe hope the classification above is a worthwhile contribution that will help to clear up the muddywaters of morphological computation and at the same time facilitate progress in very practical termswhen designing systems. One additional caveat, however, may lie in the fact that a clear separation of



the tasks is not always possible. Whereas in artificial systems like robots there is classically a completeseparation of the controller (which is a computer ) and the body (also called plant—a physicaldevice), in biological systems the brain and the body are so tightly intertwined that this separationis not feasible. The central and the peripheral neural system are often thought of as processinginformation or as computing. For example, Christoph Koch begins his classic book Biophysics ofComputation with the remark: “The brain computes! This is accepted as a truism by the majorityof neuroscientists engaged in discovering the principles employed in the design and operation ofthe nervous system” [41, p. 1]. This seems to be a prevalent framework at this point, and we willnot question it here. However, the non-neural tissue has many more complex functionalities thanbeing just a muscle-tendon system that is commanded by the brain. Rieffel et al. [69] note that apartfrom transmitting mechanical properties such as power or balance, the physical forces and displace-ments are also involved in non-neural conduits of information.

We can perhaps say that the interaction of biological organisms with the environment is orches-trated through a continuum of processes that can roughly be layered from the most low-level andphysical, through reflexes and simpler circuitry in the spine, to high-level control consisting of pureinformation processing with high bandwidths in the brain. All layers are necessary, but the lower thelevel we want to understand, the less success we will have with a computational approach.

6.2 Amidst Concepts and Buzzwords: From Computation to Orchestration?Undoubtedly, there is a lot of potential in recruiting the morphology for all the tasks we have de-scribed above. Our initial worry (Section 1.2) originated from the tension we perceive within the“morphological computation” label. As we argued, morphology is absolutely essential for a widerange of behaviors that animals or machines are engaged in. However, the contribution of the mor-phology is largely non-computational. Other concepts or labels have been proposed: intelligence bymechanics [5], morphological control ([26]; but a search suggests that this seems to be a term usedoften in chemistry, referring to the shapes of different compounds), mechanical control (used indifferent fields in different contexts), anatomical computation [81], implicit intelligence (abstractof R. Fearingʼs lecture at ETH Zurich, November 2012), and morphological communication[69]. Furthermore, there is a large overlap with the notion of embodiment, which is commonly usedto capture the phenomena of our sample cases (cf. [82], [9], and [31], where some of the charac-teristic cases used here are interpreted as embodiment showcases).

Perhaps one can say that embodiment is typically associated with slightly higher-level phenom-ena, in particular how the embodied interaction with the environment affects cognition, while themorphological computation community is typically concerned with lower-level traits. Still, as wehave argued, morphological computation refers to facilitating control and preprocessing sensoryinformation (thus facilitating perception), as well as to the possibility of “true” computation per-formed by the body. This is also reflected in the list of buzzwords above—for instance, intelligenceby mechanics, morphological control, and mechanical control seem to highlight the exploitation ofmorphology to simplify a control task, whereas morphological communication seems to emphasizethe information flows in the morphology. Perhaps, trying to cover all these facets with a meaningfullabel is futile. The overarching goal is to orchestrate the morphology so that it contributes as muchas possible to all the tasks the agent needs to master. In fact, orchestration, or “orchestration ofbehavior” [63], seems to be an expression appreciated by the community,8 which does not haveany undesired connotations or legacy.

6.3 ConclusionIn our discussion of the increasingly popular notion of morphological computation, a number ofwell-known case studies that are often cited in this context were confronted with different notionsof computation. We concluded that many of the morphological computation showcases, such as

8 As witnessed in the discussion at International Workshop on Soft Robotics and Morphological Computation in Ascona, Switzerland, 2013.



passive dynamic walkers, fail to qualify as computation in other than a metaphorical sense. Threecategories were proposed: (A) morphology facilitating control, where the walkers and other creaturesthat exploit their morphology (like geckos) fall; (B) morphology facilitating perception, encompass-ing mostly case studies where body or sensor morphology aids perceptual tasks; and (C) morpho-logical computation proper, where only physical reservoir computing qualified.

In terms of applications, leaving aside computing on nonstandard hardware, the most relevantarea where exploitation of morphology is and will be the key is probably robotics, and in particularsoft robotics ([1, 62, 79] and the first issue of the Soft Robotics Journal [78]). Soft robots break thetraditional separation of control and mechanics and exploit the morphology of the body and prop-erties of materials to assist control as well as perceptual tasks (see also [51] for a recent survey ofmaterials and technologies that can facilitate this). Pfeifer et al. [63] even discuss a new industrialrevolution. Appropriate, “cheap” designs lead to simpler control structures, and eventually can leadto technology that is cheap in a monetary sense and thus more likely to affect practical applications.Yet, a lot of research in design, simulation, and fabrication is needed (see [46] for a review). Hermanset al. [29] recently proposed a machine-learning algorithm to address the first challenge—design—ifan approximate parametric model of the systemʼs dynamics and examples of desired behavior areavailable. Kurowski and von Stryk [43, p. 3771] proposed a “systematic approach to the design ofembodiment” applied to a compliant legged robot.

Finally, the area of soft robotics and morphological computation seems to be rife with differenttrading spaces [63]. As we move from the traditional engineering framework with a central controllerthat commands a “dumb” mechanical structure toward delegating more functionality (control, per-ceptual, and computational) to the body, some convenient properties will be lost. In particular, thesolutions may not be portable to other platforms anymore, as they will become dependent on theparticular morphology and environment (a passive dynamic walker is the extreme case). The versa-tility of the solutions is likely to drop as well. To some extent, the morphology itself can be used toalleviate these difficulties—if it becomes adaptive. Online changes of morphology (such as changesof stiffness or shape) thus constitute another tough technological challenge (see [51] and projectlocomorph.eu). Completely new, distributed control algorithms that rely on self-organizing proper-ties of complex bodies and local distributed control units will need to be developed (the tensegritystructure controlled by a spiking neural network [69] is a step in this direction).

The right mix and coupling of computational and morphological building blocks will alwaysdepend on the task at hand. Some subtasks may have a clear computational nature and may be simplybest implemented on a microprocessor. Some, on the other hand, may greatly profit from exploitingthe morphology; some may even disappear altogether if a different body design is employed. Thisarticle will hopefully help the practitioners to navigate in this complex landscape.

AcknowledgmentsM.H. was supported by the Swiss National Science Foundation Fellowship PBZHP2-147259 andlater by the Marie Curie Intra European Fellowship “iCub Body Schema” (625727) within the7th European Community Framework Programme. M.H. also thanks Juan Pablo Carbajal for fruitfuldiscussions and pointers to literature. V.C.M. was supported by a James Martin Research Fellowshipat Oxford. Both authors thank our two reviewers of Artificial Life for insightful comments; theEUCogIII project (FP7-ICT 269981, coordinated by VCM) for making us talk to each other;and Rolf Pfeifer for making us think about these matters.

References1. Albu-Schaffer, A., Eiberger, O., Grebenstein, M., Haddadin, S., Ott, C., Wimbock, T., Wolf, S., &

Hirzinger, G. (2008). Soft robotics. Robotics Automation Magazine, IEEE, 15(3), 20–30.

2. Autumn, K., Sitti, M., Liang, Y. A., Peattie, A. M., Hansen, W. R., Sponberg, S., Kenny, T. W., Fearing, R.,Israelachvili, J. N., & Full, R. J. (2002). Evidence for van der Waals adhesion in gecko setae. Proceedings of theNational Academy of Sciences of the U.S.A., 99(19), 12252–12256.

3. Bajcsy, R. (1988). Active perception. Proceedings of the IEEE, 76(8), 966–1005.



4. Beer, R. D. (2003). The dynamics of active categorical perception in an evolved model agent. AdaptiveBehavior, 11(4), 209–243.

5. Blickhan, R., Seyfarth, A., Geyer, H., Grimmer, S., Wagner, H., & Guenther, M. (2007). Intelligence bymechanics. Transactions of the Royal Society, London A, 365, 199–220.

6. Boyer, F., & Lebastard, V. (2012). Exploration of objects by an underwater robot with electric sense.In T. Prescott, N. Lepora, A. Mura, & P. Verschure (Eds.), Biomimetic and biohybrid systems (pp. 50–61).Berlin: Springer.

7. Boyer, F., Lebastard, V., Chevallereau, C., Mintchev, S., & Stefanini, C. (2015). Underwater navigationbased on passive electric sense: New perspectives for underwater docking. International Journal of RoboticsResearch, 34(9), 1228–1250.

8. Brown, E., Rodenberg, N., Amend, J., Mozeika, A., Steltz, E., Zakin, M. R., Lipson, H., & Jaeger, H. M.(2010). Universal robotic gripper based on the jamming of granular material. Proceedings of the NationalAcademy of Sciences of the U.S.A., 107(44), 18809–18814.

9. Calvo-Garzón, P., & Gomila, A. (Eds.) (2008). Handbook of cognitive science: An embodied approach. Amsterdam:Elsevier.

10. Caluwaerts, K., & Schrauwen, B. (2011). The body as a reservoir: Locomotion and sensing withlinear feedback. In H. Sumioka (Ed.), 2nd International Conference on Morphological Computation(ICMC2011) (pp. 100–103).

11. Caluwaerts, K., DʼHaene, M., Verstraeten, D., & Schrauwen, B. (2013). Locomotion without a brain:Physical reservoir computing in tensegrity structures. Artificial Life, 19(1), 35–66.

12. Carbajal, J. P. (2012). Harnessing nonlinearities: Behaviour generation from natural dynamics. PhD thesis, Universityof Zurich.

13. Church, A. (1936). An unsolvable problem in elementary number theory. The American Journal of Mathematics,58, 345–363.

14. Collins, S., Ruina, A., Tedrake, R., & Wisse, M. (2005). Efficient bipedal robots based on passive dynamicwalkers. Science, 307, 1082–1085.

15. Collins, S. H., Wisse, M., & Ruina, A. (2001). A three-dimensional passive-dynamic walking robot with twolegs and knees. The International Journal of Robotics Research, 20(7), 607–615.

16. Crutchfield, J., Ditto, W., & Sinha, S. (2010). Introduction to focus issue. Intrinsic and designedcomputation: Information processing in dynamical systems—beyond the digital hegemony. Chaos,20(037101), 1–6.

17. Culha, U., Nurzaman, S. G., Clemens, F., & Iida, F. (2014). SVAS(3): Strain vector aided sensorization ofsoft structures. Sensors, 14(7), 12748–12770.

18. Dambre, J., Verstraeten, D., Schrauwen, B., & Massar, S. (2012). Information processing capacity of dynamicalsystems. Nature Scientific Reports 2, 514.

19. Degrave, J., Caluwaerts, K., Dambre, J., & Wyffels, F. (2015). Developing an embodied gait on a compliantquadrupedal robot. In J. Zhang & A. Knoll (Eds.), Intelligent robots and systems (IROS), 2015 IEEE/RSJInternational Conference (pp. 4486–4491). New York: IEEE.

20. Dodig-Crnkovic, G. (2011). Significance of models of computation, from Turing model to naturalcomputation. Minds and Machines, 21(2), 301–322.

21. Dodig-Crnkovic, G., & Müller, V. C. (2011). A dialogue concerning two world systems: Info-computational vs. mechanistic. In G. Dodig-Crnkovic & M. Burgin (Eds.), Information and computation: Essayson scientific and philosophical understanding of foundations of information and computation (pp. 149–184). Boston:World Scientific.

22. Fernando, C., & Sojakka, S. (2003). Pattern recognition in a bucket. In W. Banzhaf, J. Ziegler, T. Christaller,P. Dittrich, & J. Kim (Eds.), Advances in artificial life (pp. 588–597). Heidelberg: Springer.

23. Floreano, D., Pericet-Camara, R., Viollet, S., Ruffier, F., Brückner, A., Leitel, R., Buss, W., Menouni, M.,Expert, F., Juston, R. Dobrzynski, M. K., LʼEplattenier, G., Reckenwald, F., Mallot, H. A., & Franceschini,N. (2013). Miniature curved artificial compound eyes. Proceedings of the National Academy of Sciences of the U.S.A.,110(23), 9267–9272.

24. Floridi, L. (2004). Open problems in the philosophy of information. Metaphilosophy, 35(4), 554–582.



25. Franceschini, N., Pichon, J., & Blanes, C. (1992). From insect vision to robot vision. Transactions of the RoyalSociety, London B, 337, 283–294.

26. Füchslin, R., Dzyakanchuk, A., Flumini, D., Hauser, H., Hunt, K., Luchsinger, R., Reller, B., Scheidegger,S., & Walker, R. (2013). Morphological computation and morphological control: Steps towards a formaltheory and applications. Artificial Life, 19(1), 9–34.

27. Hauser, H., Ijspeert, A., Füchslin, R., Pfeifer, R., & Maass, W. (2011). Towards a theoretical foundation formorphological computation with compliant bodies. Biological Cybernetics, 105(5), 355–370.

28. Hauser, H., Ijspeert, A., Füchslin, R., Pfeifer, R., & Maass, W. (2012). The role of feedback inmorphological computation with compliant bodies. Biological Cybernetics, 106(10), 595–613.

29. Hermans, M., Schrauwen, B., Bienstman, P., & Dambre, J. (2014). Automated design of complex dynamicsystems. PLoS ONE, 9(1), e86696.

30. Hoffmann, M., & Müller, V. C. (forthcoming). Simple or complex bodies? Trade-offs in exploiting bodymorphology for control. In G. Dodig-Crnkovic & R. Giovagnoli (Eds.), Representation of reality: Humans,animals and machines. Berlin: Springer. Available at http://www.springer.com/qp/book/9783319437828.

31. Hoffmann, M., & Pfeifer, R. (2011). The implications of embodiment for behavior and cognition: Animaland robotic case studies. In W. Tschacher & C. Bergomi (Eds.), The implications of embodiment: Cognition andcommunication (pp. 31–58). Exeter: Imprint Academic.

32. Horsman, C., Stepney, S., Wagner, R., & Kendon, V. (2014). When does a physical system compute?Proceedings of the Royal Society A, 470(2169), 20140182, 1–25.

33. Iida, F., Gómez, G., & Pfeifer, R. (2005). Exploiting body dynamics for controlling a running quadrupedrobot. In B. Hannaford (Ed.), Proceedings of the 12th International Conference on Advanced Robotics (ICAR05)(pp. 229–235). New York: IEEE.

34. Ikemoto, S., Nishigori, Y., & Hosoda, K. (2012). Advantages of flexible musculoskeletal robot structure insensory acquisition. Artificial Life and Robotics, 17(1), 63–69.

35. Ilievski, F., Mazzeo, A. D., Shepherd, R. F., Chen, X., & Whitesides, G. M. (2011). Soft robotics forchemists. Angewandte Chemie, 123(8), 1930–1935.

36. Jäger, H. (2002). Tutorial on training recurrent neural networks, covering BPPT, RTRL, EKF and the “echo statenetwork” approach. GMD Report 159, German National Research Center for Technology. Available atminds.jacobs-university.de/sites/default/files/uploads/papers/ESNTutorialRev.pdf (accessed May 2016).

37. Jäger, H., & Haas, H. (2004). Harnessing nonlinearity: Predicting chaotic systems and saving energy inwireless communication. Science, 304(5667), 78–80.

38. Johnson, C., Philippides, A., & Husbands, P. (2016). Active shape discrimination with compliant bodies asreservoir computers. Artificial Life, 22(2), 241–268.

39. Kim, S., Spenko, M., Trujillo, S., Heyneman, B., Mattoli, V., & Cutkosky, M. R. (2007). Whole body adhesion:Hierarchical, directional and distributed control of adhesive forces for a climbing robot. In P. Fiorini (Ed.),Robotics and Automation, 2007 IEEE International Conference (pp. 1268–1273). New York: IEEE Press.

40. Kleene, S. C. (1952). Introduction to metamathematics. Amsterdam: North Holland.

41. Koch, C. (1999). Biophysics of computation: Information processing in single neurons. New York: Oxford University Press.

42. Koditschek, D. E., Full, R. J., & Buehler, M. (2004). Mechanical aspects of legged locomotion control.Arthropod structure and development, 33, 251–272.

43. Kurowski, S., & von Stryk, O. (2015). A systematic approach to the design of embodiment with applicationto bio-inspired compliant legged robots. In J. Zhang & A. Knoll (Eds.), Intelligent Robots and Systems (IROS),2015 IEEE/RSJ International Conference (pp. 3771–3778). New York: IEEE.

44. Larger, L., Soriano, M., Brunner, D., Appeltant, L., Gutierrez, J., Pesquera, L., Mirasso, C., & Fischer, I.(2012). Photonic information processing beyond Turing: An optoelectronic implementation of reservoircomputing. Optics Express, 20(3), 3241–3249.

45. Lind, D. A. (1995). An introduction to symbolic dynamics and coding. Cambridge, UK: Cambridge UniversityPress.

46. Lipson, H. (2013). Challenges and opportunities for design, simulation, and fabrication of soft robots.Soft Robotics, 1(1), 21–27.



47. Lucosevicus, M., & Jaeger, H. (2009). Survey: Reservoir computing approaches to current neural networktraining. Computer Science Review, 3(3), 127–149.

48. Lungarella, M., & Sporns, O. (2006). Mapping information flow in sensorimotor networks. PLoSComputational Biology, 2, 1301–1312.

49. Maass, W., Natschläger, T., & Markram, H. (2002). Real-time computing without stable states: A newframework for neural computation based on perturbations. Neural Computation, 14(11), 2531–2560.

50. Mammano, F., & Nobili, R. (1993). Biophysics of the cochlea: Linear approximation. The Journal of theAcoustical Society of America, 93, 3320.

51. McEvoy, M., & Correll, N. (2015). Materials that couple sensing, actuation, computation, andcommunication. Science, 347(6228), 1261689.

52. McGeer, T. (1990). Passive dynamic walking. The International Journal of Robotics Research, 9(2), 62–82.

53. Moore, A., Ober-Blöbaum, S., & Marsden, J. E. (2012). Trajectory design combining invariant manifoldswith discrete mechanics and optimal control. Journal of Guidance, Control, and Dynamics, 35(5), 1507–1525.

54. Müller, V. C. (2009). Symbol grounding in computational systems: A paradox of intentions. Minds andMachines, 19(4), 529–541.

55. Müller, V. C. (2011). On the possibilities of hypercomputing supertasks. Minds and Machines, 21(1), 83–96.

56. Müller, V. C. (2014). Pancomputationalism: Theory or metaphor? In R. Hagengruber & U. Riss (Eds.),Philosophy, computing and information science (pp. 213–221). London: Pickering & Chattoo.

57. Nakajima, K., Hauser, H., Kang, R., Guglielmino, E., Caldwell, D. G., & Pfeifer, R. (2013). A soft body asa reservoir: Case studies in a dynamic model of octopus-inspired soft robotic arm. Frontiers in ComputationalNeuroscience, 7(91). doi:10.3389/fncom.2013.00091.

58. Nakajima, K., Hauser, H., Li, T., & Pfeifer, R. (2015). Information processing via physical soft body. NatureScientific Reports, 5(10487). doi:10.1038/srep10487.

59. Nurzaman, S., Yu, X., Kim, Y., & Iida, F. (2014). Guided self-organization in a dynamic embodied systembased on attractor selection mechanism. Entropy, 16(5), 2592–2610.

60. Paquot, Y., Duport, F., Smerieri, A., Dambre, J., Schrauwen, B., Haelterman, M., & Massar, S. (2012).Optoelectronic reservoir computing. Nature Scientific Reports, 2(287). doi:10.1038/srep00287.

61. Pfeifer, R., & Bongard, J. C. (2007). How the body shapes the way we think: A new view of intelligence. Cambridge,MA: MIT Press.

62. Pfeifer, R., Lungarella, M., & Iida, F. (2012). The challenges ahead for bio-inspired ‘soft’ robotics.Communications of the ACM, 55(11), 76–87.

63. Pfeifer, R., Marques, H. G., & Iida, F. (2013). Soft robotics: The next generation of intelligent machines.In Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI) (pp. 5–11).

64. Pearson, M. J., Mitchinson, B., Sullivan, J. C., Pipe, A. G., & Prescott, T. J. (2011). Biomimetic vibrissalsensing for robots. Philosophical Transactions of the Royal Society B, 366(1581), 3085–3096.

65. Piccinini, G. (2003). Computations and computers in the sciences of mind and brain. PhD thesis, University ofPittsburgh. Available at http://etd.library.pitt.edu/ETD/available/etd-08132003-155121/.

66. Piccinini, G. (2011). The physical Church-Turing thesis: Modest or bold? British Journal for the Philosophy ofScience, 62(4), 733–769.

67. Prescott, T. J., Diamond, M. E., & Wing, A. M. (2011). Active touch sensing. Philosophical Transactions of theRoyal Society of London B: Biological Sciences, 366(1581), 2989–2995.

68. Prescott, T. J., Pearson, M. J., Mitchinson, B., Sullivan, J. C. W., & Pipe, A. G. (2009). Whisking with robotsfrom rat vibrissae to biomimetic technology for active touch. IEEE Robotics and Automation Magazine, 16(3),42–50.

69. Rieffel, J. A., Valero-Cuevas, F. J., & Lipson, H. (2010). Morphological communication: Exploiting coupleddynamics in a complex mechanical structure to achieve locomotion. Journal of the Royal Society Interface,7(45), 613–621.

70. Rückert, E., & Neumann, G. (2013). Stochastic optimal control methods for investigating the power ofmorphological computation. Artificial Life, 19, 115–131.



71. Saranli, U., Buehler, M., & Koditschek, D. (2001). RHex: A simple and highly mobile hexapod robot.International Journal of Robotics Research, 20, 616–631.

72. Smerieri, A., Duport, F., Paquot, Y., Schrauwen, B., Haelterman, M., & Massar, S. (2012). Analog readoutfor optical reservoir computers. Available at http://arxiv.org/abs/1209.3129 (accessed May 2016).

73. Song, Y. M., Xie, Y., Malyarchuk, V., Xiao, J., Jung, I., Choi, K.-J., Liu, Z., Park, H., Lu, C., Kim, R.-H.,Li, R., Crozier, K. B., Huang, Y., & Rogers, J. A. (2013). Digital cameras with designs inspired by thearthropod eye. Nature, 497(4774), 95–99.

74. Sornkarn, N., Howard, M., & Nanayakkara, T. (2014). Internal impedance control helps information gain inembodied perception. In Y. Liu (Ed.), Proceedings of the IEEE International Conference on Robotics and Automation(ICRA) (pp. 6685–6690). New York: IEEE.

75. Swapnil, P., DeForest, M., Edward, J. B., & Matthew, A. R. (2015). Effects of sensilla morphology onmechanosensory sensitivity in the crayfish. Bioinspiration & Biomimetics, 10(3), 036006.

76. Taga, G. (1995). A model of the neuro-musculo-skeletal system for human locomotion I. Emergence ofbasic gait. Biological Cybernetics, 73, 97–111.

77. Taga, G. (1995). A model of the neuro-musculo-skeletal system for human locomotion II. Real-timeadaptability under various constraints. Biological Cybernetics, 73, 113–121.

78. Trimmer, B. (2013). A journal of soft robotics: Why now? Soft Robotics, 1(1), 1–4.

79. Trivedi, D., Rahn, C., Kier, W., & Walker, I. (2008). Soft robotics: Biological inspiration, state of the art,and future research. Applied Bionics and Biomechanics, 5(3), 99–117.

80. Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungsproblem.Proceedings of the London Mathematical Society, 42, 230–256.

81. Valero-Cuevas, F. J., Yi, J.-W., Brown, D., McNamara, R. V., Paul, C., & Lipson, H. (2007). The tendonnetwork of the fingers performs anatomical computation at a macroscopic scale. IEEE Transactions onBiomedical Engineering, 54(6), 1161–1166.

82. Varela, F. J., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience.Cambridge, MA: MIT Press.

83. Wiesner, K., & Crutchfield, J. P. (2008). Computation in finitary stochastic and quantum processes.28.04.2008. Available at http://arxiv.org/abs/quant-ph/0608206 (accessed May 2016).

84. Zenil, H. (2013). A behavioural foundation for natural computing and a programmability test. In G. Dodig-Crnkovic & R. Giovagnoli (Eds.), Computing nature: Turing centenary perspective (pp. 87–113). Berlin,Heidelberg: Springer.



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